【作者】 李娜；

【导师】 彭海军；

【作者基本信息】 大连理工大学 ， 工程力学， 2024， 博士

【摘要】 柔性结构由于质量轻、灵活度高、易于设计成各种形状等优点,已经广泛被应用于航空航天、医疗仿真、机器人、海洋工程等领域。目前,针对柔性结构的建模方法主要基于力学方法。受几何大变形和材料非线性影响,纯力学方法建模复杂,计算成本大,不利于实时仿真,并且在一定程度上限制了柔性结构相关控制器的设计。近几年,柔性结构的快速仿真在计算机图形学领域逐渐成为研究热点。然而,在计算机图形学中柔性结构建模主要基于几何模型,并不涉及弹性模量、泊松比等力学物理量,因此无法与实际材料相结合。本文基于计算机图形学中的扩展位置动力学(Extended Position based Dynamics,XPBD),将其与连续介质力学相结合,提出一种可以反映结构力学特性的XPBD方法。依照单元类型,本文将柔性结构分为平面柔性结构和三维柔性结构进行专题研究,并根据不同结构面临的问题提出相应的控制设计方案。因此,本博士论文主要的研究内容如下:(1)针对平面柔性结构动力学建模问题,本文采用XPBD描述平面柔性结构模型,并推导应变约束对XPBD进行改进。传统XPBD采用基于位置关系的几何约束构建模型,无法反映实际的结构变形,无法对结构进行力学分析。为了克服这种情况,本文基于应变能函数推导出应变约束对几何约束进行替换。考虑到平面柔性结构在展开过程会涉及接触碰撞,传统暴力搜索会降低仿真效率,因此本文利用包围盒算法和连续碰撞检测构建了柔性结构快速仿真框架。为了验证所提方法的可行性,本文对充气气囊进行仿真,并与基于张力场理论的互补共旋有限元法进行对比,结果表明:所提方法既可以实现气囊的力学分析又能准确预测气囊的褶皱区域。另外,本文对航天充气天线进行动力学仿真,验证了所提方法对复杂平面柔性结构的可行性。(2)针对平面柔性结构充气展开问题,本文提出一种充气展开最优控制方法。充气管作为平面柔性结构的一种典型支撑框架,在充气时存在各个腔体压力不均并且逐个展开的现象。因此,本文根据折痕对充气管进行腔体划分,采用控制体积法建立每个腔体的气体状态微分方程,通过积分计算各腔体压力。之后,考虑到充气速率较大时虽然平面柔性结构可以快速展开,但展开过程并不稳定,完全展开后存在残余振荡;而充气速率较小时,虽然可以减小平面柔性结构展开后的振荡幅度,却延长了展开时间。因此,本文基于瞬时最优控制方法提出变充气速率概念,通过推导充气速率的微分控制方程实现对每一时刻充气速率的控制,使平面柔性结构在展开过程中可以兼顾展开时间和稳定性。本文分别对Z字形四折充气管和航天遮阳罩模型进行仿真与控制,结果表明:所提方法既可以使平面柔性结构快速展开,又可以防止结构完全展开后出现较大的振幅,同时也为恒定充气速率的选择提供参考。(3)针对三维柔性结构建模仿真问题,本文将XPBD建模扩展至三维柔性结构,并在此基础上提出一种新的非线性模型降阶方法。考虑到三维柔性结构受材料和驱动影响,运动过程往往涉及几何非线性和材料非线性,仿真时间成本大,本文提出一种采用线性模态和模态导数组成基底进行降阶的非线性降阶算法。首先,基于结构初始状态推导出线性模态对广义坐标的一阶导数(又称模态导数);之后,通过线性模态和其对应的模态导数构建降阶矩阵;最后,考虑到内力阵的每一项均为多项式形式,在降阶过程对相同项进行系数合并,减少迭代更新中的计算量,提高仿真效率。本文通过对绳驱和气动软体机器人仿真并与实验相比,结果表明:所提方法可以实现三维柔性结构实时仿真,并为三维柔性结构的设计验证以及结构分析提供一定的参考。(4)针对三维柔性结构轨迹跟踪问题,本文基于XPBD降阶模型提出轨迹跟踪瞬时最优控制方法。首先,将绳索驱动以几何约束引入,建立基于降阶模型的软体机器人轨迹跟踪控制问题;之后,根据瞬时最优控制原理,将受控等式约束方程在连续时间域上进行离散;然后,利用牛顿迭代求解每个瞬时目标函数的最小值,并得到该瞬时的最优控制输入;最后,将瞬时最优控制代入动力学模型中更新状态变量,并根据收敛准则判断是否进入下一时间步。同时,为了进一步验证所提方法的可行性和稳定性,借助视觉运动捕获系统、伺服电机等硬件设备搭建轨迹跟踪闭环控制实验平台,对3D打印制造的绳驱柔性机械臂进行轨迹跟踪控制实验。结果表明:所提方法可以较好地控制机械臂末端实时跟踪圆弧、折线组成的各种复杂轨迹。另外,在跟踪过程中即使对柔性机械臂施加扰动或者在末端添加砝码,所提方法依旧可以控制柔性机械臂完成轨迹跟踪任务。更多还原

【Abstract】 Flexible structures have been widely used in aerospace,medical simulation,robotics,ocean engineering and other fields because of their light weight,high flexibility,and easy to design into various shapes.At present,the modeling methods for flexible structures are mainly based on mechanics.However,due to the influence of large geometric deformation and material nonlinearity,the modeling of flexible structures is complex and the calculation cost is large,which is not conducive to real-time simulation,and the relevant controller design is limited.In recent years,real-time simulation of flexible structures has gradually become a hotspot in the field of computer graphics.However,most of the modeling methods are based on geometrical relationship,which cannot be combined with real materials.The Extended Position based Dynamics(XPBD)which is popular in computer graphics,is improved in the present paper and can simulate flexible structures in real-time.According to the structure type,the flexible structures are divided into planar flexible structures and three-dimensional flexible structures,the corresponding controller are designed.The feasibility of the proposed method is verified by numerical simulations and experiments.Therefore,the main contents of this doctoral dissertation are as follows:(1)For the dynamic modeling of planar flexible structure,XPBD is used to model the planar flexible structures,and strain constraints are derived to improve XPBD.The traditional XPBD usually uses geometric constraints to construct the model,which cannot be combined with real materials or reflect the mechanical characteristic,so,this paper derives the strain constraints based on the strain energy function to improve it.In addition,considering that collision may occur during the deployment process of planar flexible structures,a fast simulation framework of flexible structures is constructed by using bounding box algorithm and continuous collision detection.In order to verify the feasibility and accuracy of the proposed method,an inflatable airbag is simulated,the results are compared with the complementary co-rotation finite element method which is based on the tension field theory.The simulation results show that the proposed method is simple,stable,and can accurately predict the displacement,stress and wrinkled area of the airbag.In addition,aerospace inflatable antenna is also simulated,the results verify the feasibility of the proposed method for complex planar flexible structures.(2)For the inflation deployment of planar flexible structures,an optimal control method is proposed based on XPBD model.As a typical braced frame of planar flexible structures,each cavity of the inflatable tube has different pressure and deploys one by one.Therefore,the inflatable tube is divided into cavities according to the crease,the gas state equations of each cavity is established by the control volume method,and the pressure of each cavity is calculated by integration.After that,considering that the flexible planar structure can deploy rapidly when the inflation rate is large,but the process is not stable,and the vibration often occurs after completely deploying.When the inflation rate is small,the vibration amplitude declines,but the deployment time prolongs.Therefore,based on the instantaneous control method,the concept of variable inflation rate is proposed in the present paper.By constructing the differential governing equation of inflation rate,the deployment time and stability of planar flexible structures are controlled.The simulation results show that the proposed method can effectively balance the deployment time and stability,which can not only make the flexible structures deploy quickly,but also prevent large amplitude after completely deploying.At the same time,it also provides the reference for the selection of constant inflation rate.(3)For the simulation efficiency of three-dimensional(3D)flexible structures,XPBD is extended to model 3D flexible structures,and a novel nonlinear model order reduction method is proposed.Considering that 3D flexible structures are influenced by materials and actuations,the deformations often involve geometric nonlinearity and material nonlinearity,and the simulation time cost is large.To improve the computational efficiency,the present paper derives the modal derivative to reduce the order of the model.Firstly,the modal derivatives are derived based on the linear modes.Then,the reduced-order matrix is constructed by the linear modes and their corresponding modal derivatives.Finally,considering that each item of the internal force matrix and stiffness matrix can be expressed as a polynomial form,the coefficients can be merged during the model order reduction process,it makes the amount of calculation decline in the iterative update and improves the simulation efficiency.The simulation results of cable-driven and pneumatic flexible robots show that the proposed method not only realize the real-time simulation,but also provide a certain reference for 3D flexible structures to verify the design structural analysis.(4)For the trajectory tracking problem of 3D flexible structures,an instantaneous optimal controller for trajectory tracking based on reduced-order XPBD model is proposed.Firstly,the dynamic equations of 3D flexible structures are established by reduced-order XPBD.The cable drive is introduced with geometric constraints,and the problem of the trajectory tracking control is established.Then,according to the instantaneous optimal control,the equality constraint equations are discretized on continuous coordinates.Then,the minimum value of each instantaneous objective function is solved by Newton iteration,and the instantaneous optimal control input is obtained.Finally,the optimal control input is substituted into the dynamic model to update the state variables,and the convergence criterion is used to decide whether to enter the next time step.At the same time,in order to further verify the feasibility and stability of the proposed method,a closed-loop control experimental platform is built.The experimental platform consists of visual motion capture system,servo motor,3D printing,etc.The trajectory tracking experiment of the cable-driven flexible manipulator is carried out.The results show that the proposed method can well control the end of the manipulator to track various complex trajectories in real time,which are composed of circular arcs and polylines.In addition,in the tracking process.In addition,even if the flexible manipulator is perturbed or the weight is added to the end,the proposed method can still complete the trajectory tracking task,which shows the robustness of the proposed method.更多还原